# Laurent Series

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## Laurent series

[lȯ′ränz ‚sir·ēz]## Laurent Series

a series of the form

(*)

that is, a series of positive and negative powers of the difference z — *a* (where z, *a*, and the coefficients of the series are complex numbers). Here, the nonnegative powers form an ordinary power series that converges, in general, within a circle with center *a* and radius R ≤ ∞. The remaining terms form a series that converges, in general, outside a circle with the same center but with radius *r* ≥ 0. If *r* < *R*, then the series (*) converges in the annulus *r* < ǀz — *a* < *R* to an analytic function of the complex variable z.

Although series of the form (*) were already known to L. Euler (1748), they were named after P. Laurent, who in 1843 proved that every function of a complex variable, single-valued and analytic in the annulus *r* < |z — | < *R*, can be expanded in this annulus in this type of series (Laurent theorem). The theorem, however, was obtained somewhat earlier by K. Weierstrass in a work that remained unpublished until 1894.